## On a $$\Phi$$-Kirchhoff multivalued problem with critical growth in an Orlicz-Sobolev space.(English)Zbl 1304.35254

Summary: This paper is concerned with the multiplicity of nontrivial solutions in an Orlicz-Sobolev space for a nonlocal problem with critical growth, involving $$N$$-functions and theory of locally Lipschitz continuous functionals. More precisely, in this paper, we study a result of multiplicity to the following multivalued elliptic problem: $\begin{cases} -M \left (\displaystyle \int_\varOmega \varPhi(|\nabla u|){\mathrm {d}}x \right ) \Delta_\varPhi u \in \partial F(\cdot,u)+\alpha h(u) \quad \text{in }\varOmega, \\ u \in W^{1}_0L_\varPhi(\varOmega), \end{cases}$ where $$\varOmega \subset \mathbb{R}^N$$ is a bounded smooth domain, $$N \geqslant 3$$, $$M$$ is a continuous function, $$\varPhi$$ is an $$N$$-function, $$h$$ is an odd increasing homeomorphism from $$\mathbb{R}$$ to $$\mathbb{R}$$, $$\alpha$$ is positive parameter, $$\Delta_\varPhi$$ is the corresponding $$\varPhi$$-Laplacian and $$\partial F(\cdot,t)$$ stands for Clarke generalized of a function $$F$$ linked with critical growth. We use genus theory to obtain the main result.

### MSC:

 35J30 Higher-order elliptic equations
Full Text: